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Thursday, January 31, 2008

Summer Holiday

A little animal friend told me that Betsy Devine is currently blogging from New Zealand! She has an odd collection of posts on the North Island, the latest entitledIn some alternate universe we are all surfer dudes.

The presence of many Americans in NZ at this time of year does not go unnoticed by those of us who have to run around madly carrying salads and glasses of fine South Island pinot noirs to the tables outside in the sun. If any visiting physics bods want a really enthusiastic and cheap travel guide around the Southern Alps, then just give me a bell!

Monday, January 28, 2008

Swagger Again

With the surreals, this relation could extend all the way to $\omega$, resulting in the nonsensical equality

$\frac{1}{6} \omega (\omega + 1)(2 \omega + 1) = \zeta (-2)$

in contrast to the usual definition, where $\zeta (-2)$ is zero. Moreover, it needs to be zero to cancel the pole of $\Gamma (-2)$ in the functional relation defining $\zeta (3)$. This suggests that, paradoxically, $\Gamma (-2)$ should be expressed in terms of the infinitesimal $\omega^{-1} = \varepsilon$. Perhaps we got $\zeta$ and $\Gamma$ mixed up!

Now consider the interesting number $\zeta (2) = \frac{\pi^{2}}{6}$, defined using $\zeta (-1) = \frac{1}{2} \omega (\omega + 1)$. The $\Gamma$ function is conveniently infinite again, usually in order to balance a zero from the sin factor in the functional relation. It seems necessary to balance an awful lot of zeros and infinities just to define the $\zeta$ function. A surreal zeta function may distinguish different zeros with polynomials in $\varepsilon$. Wouldn't that be fun?

Saturday, January 26, 2008

Categorical Aside

After a few lectures of basic category theory, people often become quite enthusiastic about discussing their favourite examples of objects and morphisms, as if recognising a category for what it is will be magically useful somehow. After a few more lectures their enthusiasm is usually dampened by the obtuseness of it all, and the realisation that just lumping things into categories doesn't really get one anywhere.

And then, after learning some more tricks, there is a tendency to apply these tricks to the same old examples that came up in the first place. For example, we often discuss the category of (finite dimensional) vector spaces over a field $\mathbb{F}$, where it doesn't really matter what $\mathbb{F}$ is, because the only structure given to the category is the basic properties of a vector space, and its ability to be tensored with other spaces. So we might as well be discussing the category of vector spaces over $\mathbb{F}_{2}$, the field with two elements.

In M Theory, lumping everything into an arbitrary well-known category (or functor category) is analogous to deciding that path integrals for quantum gravity should rely on classical geometry: it amounts to making a ridiculously unacceptable assumption about Nature's way of doing geometry. The category theory itself should provide the geometry. Alternatives tend to be tricky, and require delving into axiom systems, or obscure logic and philosophy, but calculating is eventually meant to be easy!

Friday, January 25, 2008

Pretty Preons

M Theory Lesson 150

Quite a while back, kneemo recommended a paper by Duff and Ferrara on the two way entanglement of three qutrits. The paper actually begins by looking at qubits associated to $4D$ stringy black holes, and in particular the use of a hyperdeterminant to express the entropy.

This hyperdeterminant is invariant under the U triality, which is a kind of three (spatial) dimensional analogue of the two dimensional duality currently generating much interest. Thus it is no surprise that when they move on to seven qubits and tripartite entanglement (giving seven lines with three nodes on the Fano plane) we start to see the familiar circulants, this time associated with $E(7)$, namely the matrix

Observe that this circulant is basically the $7 \times 7$ circulant for the Hamming code, with $1$ added to each entry, and indeed this circulant is associated to the Fano plane and seven bits of information. Moreover, an $E(8)$ interpretation has the advantage of agreeing with the 3 Time interpretation of the spatial dimensions, at least in the context of M Theory.

By considering the entries of the matrix above to be qutrit elements, $A_{ij} \in \{ 0,1,2 \} = \mathbb{F}_{3}$, we see that the addition of $1$ to each entry again yields a circulant, which is twice the complement of the Hamming circulant. And finally, yet another addition of unit entries returns the matrix to the Hamming circulant. Thus a triality is made manifest by the root vector circulants.

Duff and Ferrara point out that the question of real forms for $E(7)$ is not really important in this context, since the coefficients defining the state are allowed to be complex. Hmm. This also sounds like something that came up recently.

Wednesday, January 23, 2008

Brave New World

I joined Facebook just to look at a few photos, but then momentarily found myself drowning in the temptations of procrastination, so I joined the groupThe Petition for Alexander Grothendieck to Return From Exile. A local paper ran a feature last week on the Evil of Facebook, but the evils discussed sounded more like a list of shallow 20th century social constructs, ever present on the web due to the conditioning of its participants.

But a Brave New World could lead to an even braver New World, if we so desired. We can create the benevolent Big Sister, one who looks back on the 20th century respectfully, but in dismay. For instance, much funding for such sites clearly comes from advertising revenue. How can we remove advertising from the forums of the future? Personally, I don't want to discuss life, the universe and everything whilst being bombarded by pictures of women wearing underwear that doesn't fit properly. I went into one of those popular lingerie stores once and was disgusted to find that none of the expensive items was well made, or fitted. That implies that their revenue is generated entirely by trend value, gladly paid by women, many of whom are technically living below the poverty line and should be spending the money on more fresh fruit and vegetables.

Associativity

It is easy enough to invent associative algebraic structures that quickly lead to associated non-associative structures. Unfortunately, ordinary numbers are almost always used as models in the sense that addition is assumed to be commutative. Let us consider a system with two binary operations, by convention called addition and multiplication, neither of which is commutative. Let us assume that scalars are associative and distributive, so that

$x(y + z) = xy + xz \neq xz + xy = x(z + y)$

What happens with $2 \times 2$ matrices over these scalars? Matrix multiplication is well defined by the usual rule, but one must be careful about ordering scalars. In a triple product of matrices $ABC$, associativity is lost, because the first element of the product $A(BC)$ is given by

Monday, January 21, 2008

M Theory Lesson 149

Now the hexagon that runs through the three discs of the pants in lesson 144, and along the real axis on the Riemann sphere, is just like the hexagon (cyclohedron) from yesterday, because three edges are labelled by single chord hexagons (the squares of the associahedron in 3 dimensions) and the other three edges by edges in the associahedron which link the squares. The two vertices of the associahedron which do not appear in the circuit correspond to the two vertices of the trivalent trees drawn on the pair of pants.

So this hexagon is a real dimension shifter! Previously, the three squares were associated with three faces of the mass generation cube. The completion of the cube is now seen as a pairing between the two triangular circuits of the hexagon, denoted respectively by 1-circulant and 2-circulant matrices in the Fourier transform.

Sunday, January 20, 2008

M Theory Lesson 148

Another interesting sequence of polytopes is the cyclohedra, but these are thought of as a module for an operad. The $n$th cyclohedron is a $K( \pi , 1)$ space, which means that there exists a group $\pi$ such that the first homotopy group of the space is $\pi$ and all other homotopy groups are trivial. This is mentioned in one of Devadoss's classic papers, which explains the labelling on the 2-cyclohedron, namely the hexagon. Note that hexagons with chords are also used to label the three dimensional Stasheff associahedron, but here only centrally symmetric chorded polygons appear. An appendix in a paper by Markl explains how this hexagon is turned into a triangle by shrinking down three of the sides. In general, there is a process for turning cyclohedra into regular simplices. Note that the cyclohedra are sometimes named after their inventors, Bott and Taubes.

Friday, January 18, 2008

M Theory Lesson 147

First let us consider the relation on matrices defined by $A \simeq B$ if $[A,B] = \lambda I$ for a scalar $\lambda$. It is not necessarily transitive, except for triples satisfying rules of the form

$[A,B] + [B,C] - [A,C] = 0$

but it is reflexive (since $[A,A] = 0$) and symmetric (since $[A,B] = -[B,A]$). In the world of categories we think of transitivity as a triangle of arrows, but we might weaken this triangle by allowing 2-arrows, or even higher dimensional structure.

Under this equivalence, the usual Heisenberg rule $[X,P] = i \hbar$ is a kind of equivalence between position and momentum. If we exponentiate this expression we find that

$\textrm{exp}(XP)=\textrm{exp}(i \hbar) \textrm{exp}(PX)$

which naturally reminds us of the Weyl rule for the discrete Fourier transform underlying the mass matrices. Now we see that $\hbar$ naturally defines a root of unity, and there is no reason to assume it takes on a fixed value. Moreover, when the root of unity is specified by the dimension of the matrix, as is the case for the Fourier transform, the value of $\hbar$ is specified.

How Time Flies II

Schroedinger, in his famous Dublin lectures of 1944, discussed the clockwork nature of life, and he demonstrated that new laws of physics would be required to really understand the function of genes, whose mutations occur via a single quantum molecular transition between isomers and yet may be transmitted faithfully through centuries. He also discusses how organisms feed on negative entropy (order) in their environment.

...the new principle that is involved is a genuinely physical one: it is, in my opinion, nothing else than the principle of quantum theory over again... We seem to arrive at the ridiculous conclusion that the clue to the understanding of life is that it is based on a pure mechanism, a clockwork in the sense of Planck's paper [1].

Unfortunately, I can't find a copy of Planck's paper, which might be interesting to look at as a very early discussion on the differences in physical law at different scales. But observe that life, as observed on a fixed scale, requires both kind of law, that governing the large, and that governing the small.

My, How Time Flies

But much criticism proffers counterarguments along equally problematic lines: universal evolution must do such and such, or we cannot assume typical observer types, or alternative anthropic biases, as if Time were God-given to Man, as if looking back in Time from the chains of Earth was a view into a concrete jungle, fixed for an objective eternity.

The entropy of our observable universe is mostly about black holes, the observables for which we can use completely apersonal mathematics. By the same token, quantum brains are states dependent on completely apersonal mathematics, at least in the context of M Theory, albeit mathematics which we may not yet understand. However, this mundane statement hints at the appearance of life in the New Physics, and it is there. I agree with the stringers about that! The first paragraph of the New York Times article provides a glimpse of the new intuition, but in the second paragraph we get a feeling for the actual spirit of the analysis, with the words

Nobody in the field believes that this is the way things really work, however.

M Theory Lesson 146

Thinking of ordinals $n \in \mathbb{N}$ as finite sets, one notes that the prime numbers don't seem so special any more. What is so special about a set of three oranges, as opposed to a set of four oranges? Still, a composite number of oranges can be arranged into a rectangular shape of dimension equal to the number of prime factors. Primes are then single lines of oranges. At least its nice to see that building blocks for sets are geometrically one dimensional, somehow like space filling curves.

Maybe these sets are equipped with further structure. For instance, they might be the finite fields with $p$ elements. Most fields that physicists play with have the unfortunate property of having no zero divisors, unlike the interesting operator algebras studied by Carl Brannen, where it is quite possible that $\rho_{1} \rho_{2} = 0$. The number $0$ represents an experimental beam stop: the action of allowing no Stern-Gerlach particles through, which is a simple state that one's mathematics really shouldn't ignore.

In the topos Set, the empty set is the object of cardinality zero, but we are not used to breaking the empty set up into pieces. This is a clue that the classical topos set (including set theory) is not the right setting for quantum physics, although we already knew that, because all 1-toposes rely on distributive lattices. Brannen's operator algebras also have the nice feature that the requirement of idempotency (projectors are the natural way to look at quantum lattices) specifies a normalisation for any state, removing the arbitrariness of the usual picture.

Wednesday, January 16, 2008

M Theory Lesson 145

Thinking about pairs of pants, we see that the twistor moduli surfaces are made of either four pants, or three pants, or two pants, although they are not entirely glued up like the Klein surface. If we like, we can make a $3 \times 3$ matrix of such spaces, since spaces are just a kind of higher dimensional groupoid.

Saturday, January 12, 2008

M Theory Lesson 144

The 9 faced associahedron in $\mathbb{R}^{3}$ keeps popping up in M Theory. Today we'll turn it into a pair of pants, with the three discs at the boundary corresponding to the three squares of the polytope, which were occasionally marked with crossings of a trefoil knot. The numbers labelling the real axis (which are a bit hard to read) are $-1$, $0$, $\frac{1}{2}$, $1$, $2$ and $\infty$. The first image is an extension of the Riemann sphere of lesson 62 to a hexagon on the real axis. The second image is a Grothendieck ribbon graph associated to the j invariant. Note that the ribbons pass through $-1$, $\frac{1}{2}$ and $2$ on the real axis.

By splitting the ribbon into six pieces on the pair of pants, marked with a trivalent vertex on the back and front, and attaching vertices to the nodes of the projection onto the plane, we find exactly 14 vertices, six pentagons and three squares, describing the associahedron. This might just be a bit of fun, until we look at what happens when we glue four of these pants together to form a genus 3 surface. By adding vertices on the squares from each side of the gluing, the pentagons are turned into heptagons, and we get a 24 heptagon tiling of the Klein quartic. Who said operads weren't useful?

Wednesday, January 09, 2008

There in Time

Most discussions of the Fermi paradox, at least amongst scientists, involve the mundane multiplication of several numerical factors, pulled out of the air, such as a concentration of life-supporting planets in the galaxy. But as Matti Pitkanen has discussed, a quantum gravitational view of the cosmos allows a more interesting resolution of the paradox.

Let us assume, first of all, that we are not special. That is, similar types of observer are possible in galactic or planetary states that differ from ours. Secondly, as the example of Earth illustrates, let us assume that the 20th century technological state is locally short lived. Then, even restricting ourselves to classical ideas of information transmission, we should not expect to detect alien signals until our ideas of information encoding become a lot more sophisticated.

But a truly quantum gravitational view of things would say something more profound: perhaps we haven't met alien dust because it isn't Time. In the evolution of emergent cosmic time, as viewed by Earthlike observers, the cosmos must reach a threshhold of complexity before its state can accomodate the transmission of information between galactic civilisations. Thus our state of understanding itself is the barrier that one day, if we survive, we might break.

Sunday, January 06, 2008

Riemann Rekindled III

I haven't had a chance to watch GRT Lecture 19 yet, but the puzzle is to find a groupoid with cardinality $\pi$ and one with cardinality $e^e$. What fun! As it happens, I was thinking about $e^e$ last night, because

and that made me wonder about rescalings of the Riemann zeta function, such as

$F(s) = \sqrt{\frac{1}{2} (s+1) (s+2) \zeta (s) }$

in terms of which

$i = \frac{\textrm{log} (1 + 2 + 4 + 8 + \cdots)}{F(2)}$

with $F(2) = \pi$, and this looks something like a count of binary trees, with an increasing number of branches at each step. What are the higher dimensional analogues of $i$? What if we took the $s$-th root, so that $F(2n)$ was some multiple of $\pi$ for all $n \in \mathbb{N}$, just like the volumes of spheres?

Riemann Rekindled II

It's wonderful to see the GRT lectures reach the topic of degroupoidification and homology. The dual groupoidification process would take a vector space to a groupoid, which might be a one object groupoid. An instance of such a process might be the exponentiation of a Lie algebra to its Lie group. Dually, multiple logarithms are associated to many object degroupoidifications, as we see with the MZV algebras.

Now a while back the Everything Seminar set off a series of posts on categorified sums, including goodies like

$-1 = 1 + 2 + 4 + 8 + \cdots$

so we expect Euler's relation may be written in many ways, such as

$\textrm{log} (-1) = i \pi = \textrm{log} (1 + 2 + 4 + 8 + \cdots)$

which I guess is a definition of $\textrm{log} (-1)$, or maybe of $\pi$, which turns up in the Riemann zeta function for integral arguments. I wish I could play this game all day, but alas, the restaurant is busy...

Friday, January 04, 2008

Riemann Rekindled

The demise of the arxiv continues into 2008 with yet another (cough) disproof of the Riemann Hypothesis (reported by Lubos). Elementary disproofs seem popular these days. Since Connes tells us the Riemann Hypothesis is closely related to Quantum Gravity, that means Quantum Gravity must be Elementary also. Elementary in the sense of axiomatically foundational, maybe?

Yesterday we came across categorified cardinalities once again. For example, to compute the cardinality of the groupoid of finite sets we just need to sum the cardinalities of the groupoid components,

so when $s$ is a positive integer this might measure the cardinality of the sequence of products of cyclic groups $( \mathbb{Z}_{n} )^{s}$ for $n \in \mathbb{N}$. What sort of groupoid is this? It is very reminiscent of Rota's ideas on profinite combinatorics and the Riemann zeta function. Hmmm. We know that $\zeta (2) = \frac{\pi^{2}}{6}$ and so on, so the factors of $\pi$ must come from a cardinality for such a groupoid. The question is, what basic thing has (products of) cyclic automorphism groups? One possibility is oriented polygons and we already know that $n$-gons are associated with $n-3$ dimensional associahedra, and associahedra are related to the permutohedra, the vertices of which give the elements of the groups counted by $e$.

This seems like such a nice way to relate $e$ and $\pi$ and $-1 = e^{i \pi}$.

Thursday, January 03, 2008

GRT Wonderland

John Baez recommends further installments of Geometric Representation Theory. In Lecture 17 see James Dolan explain how degroupoidification is related to logos theory! If a map taking sets to the trivial category (which we think of as a -1 category) is decategorification, then an enriched version takes us from categories to categories enriched in the trivial category, which are just sets (with Boolean truth values). Imagine a whole recursion process of decategorifications!

In Lecture 17 you can also see homology and cohomology rear their Medusa's heads! These functors give a way to turn spans (the natural morphisms which have been floating around) into matrices. Then Baez's Lecture 18 looks at the example of the groupoid of finite sets (and bijections), before explaining how groupoid cardinality can be a fraction! Recall that this came up when we looked at Abel sums and counting trees, not to mention that Euler characteristics for orbifolds are secretly this kind of number! That's one of the ways we counted the number of particle generations in M theory.

Our preference for operadification and cooperadification should be viewed with these new ingredients in mind. Remember that an operad is a one object multicategory. An example of an arrow in a multicategory is a cospan diagram, which is made of two arrows with the same target. Multicategories generalise to allow arrows with an arbitrary number of inputs and outputs. Fixing our attention on the 1-operad of associahedra, recall that the coherence law dimension is associated to the number of inputs. Thus spans and cospans are naturally associated with two dimensional structures underlying duality. An instance of duality can be seen in the cardinalities for $Z_{2}$ (appearing in Lecture 18), namely 2 as a set and $\frac{1}{2}$ as a groupoid! Decategorification takes groupoids to vector spaces (or sets), and cardinality is thus reduced to an integer.

A Green Year

Check out your Conservation Profile at Earthlab. Scores tend to range from 150 to 900. The average score in the U.S. is 325. I scored 179 and have a carbon output of 2.1 tons, which I could definitely improve upon by living closer to work and eating more organic foods. Alas, poverty is the real impediment to progress.

At The Edge

This is just too funny: lest you thought the Lisi debacle was over, Garrett has contributed to The Edge New Year's essays for 2008 as an Independent Theoretical Physicist. Ooooh. His answer is one of the wittiest, by the way. CV has a summary of some essays, including a mind-bogglingly condescending one from Sean himself.

Sabbagh, the author of a book on the Riemann Hypothesis, tells us that he used to believe experts, but now he figures his guess is as good as theirs. Interesting essays on the scientific front include one by Ledoux, who explains that memory is not stored in the brain, one by Deheane on a theory of the brain being developed by Friston et al, and another by Steinhardt on taking quantum cosmology seriously. The token women include Janna Levin, who questions the assumption that the universe is infinite. Meanwhile, the journalist De Pretis discovers social processes in science.